Journal ArticleDOI
Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LP-based Approximation Algorithm
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In this article, it was shown that unless NP ⊂ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than 2 log 1-en by a reduction from the maximum label cover problem.Abstract:
We show that for every positive e > 0, unless NP ⊂ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than 2log1-en by a reduction from the maximum label cover problem. Our result also implies that Approximate Graph Isomorphism is not robust and is, in fact, 1 - e versus e hard assuming the Unique Games Conjecture.Then, we present an O(√n)-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in state-of-the-art exact algorithms.read more
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Minimizing the Union: Tight Approximations for Small Set Bipartite Vertex Expansion
TL;DR: In this paper, the Small Set Bipartite Vertex Expansion (SSBVE) problem is considered and an O(n 1/4+\varepsilon)-approximation algorithm is given for any constant ε > 0.
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Spectral Graph Matching and Regularized Quadratic Relaxations I: The Gaussian Model.
TL;DR: It is shown that GRAMPA exactly recovers the correct vertex correspondence with high probability when $\sigma = O(\frac{1}{\log n})$.
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Testing correlation of unlabeled random graphs
TL;DR: The proof of the impossibility results is an application of the conditional second-moment method, where the truncated second moment of the likelihood ratio is bound by carefully conditioning on the typical behavior of the intersection graph and taking into account the cycle structure of the induced random permutation on the edges.
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word2vec, node2vec, graph2vec, X2vec: Towards a Theory of Vector Embeddings of Structured Data.
TL;DR: A survey of vector embedding techniques can be found in this paper, where the authors propose two theoretical approaches for understanding the foundations of vector representations and draw connections between the various approaches and suggest directions for future research.
Book ChapterDOI
Efficient Rational Proofs for Space Bounded Computations
TL;DR: This work presents new protocols for the verification of space bounded polytime computations against a rational adversary, and presents protocols for randomized complexity classes, using a new composition theorem for rational proofs which is of independent interest.
References
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Assignment Problems and the Location of Economic Activities
Journal ArticleDOI
Assignment Problems and the Location of Economic Activities
Journal ArticleDOI
Proof verification and the hardness of approximation problems
TL;DR: It is proved that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P, and there exists a positive ε such that approximating the maximum clique size in an N-vertex graph to within a factor of Nε is NP-hard.
Proceedings ArticleDOI
Proof verification and hardness of approximation problems
TL;DR: Agarwal et al. as discussed by the authors showed that the MAXSNP-hard problem does not have polynomial-time approximation schemes unless P=NP, and for some epsilon > 0 the size of the maximal clique in a graph cannot be approximated within a factor of n/sup 1/ε / unless P = NP.
Journal ArticleDOI
Probabilistic checking of proofs: a new characterization of NP
Sanjeev Arora,Shmuel Safra +1 more
TL;DR: It is shown that approximating Clique and Independent Set, even in a very weak sense, is NP-hard, and the class NP contains exactly those languages for which membership proofs can be verified probabilistically in polynomial time.